294: REV. T. P. KIRKMAN ON THE ENUMERATION, DESCRIPTION, AND 



41. Construction of Knots of Eight Crossings. — For the subsolids 8 A, &c, we 

 have to draw leading flaps on G A, &c. In 6 A, which is 

 figured here, the 2-ple zoneless heteroid axis passes 

 through a and e. The only faces are 5641, 634, and 

 563, and there is one flap which is asymmetric, hav- 

 ing different edges marked / and f. The only different 

 lines that can be drawn are 



fa,fh, ca, cb, each 35 ; 

 fc and ab, each 44 ; and 

 fd,f'c, be, bd, de, dc, each 34. 



The two d's are the same point repeated in the repeated triangles 563 and 124. 



(fa) =53, 53 > (12) =53, 53? 



(fb) = 53>(12)=43; 



(ca) =53, 54>(56)=53, 53 ; (12) is fixed (art. 22) ; 



(cb) =53,44>(56)=53,44? 



(fc) =44, 43 > (12) =44, 43? 

 (/'rf) =43, 43 > (12) =43, 43? 



(bd) =43, 54>(12)=43, 53; (56) is fixed (art. 23). 

 For the rest, 



(12) leads (ab) ; and (56) leads (fc) (be), (de) and (dc). 

 We have to draw seven flaps, expecting symmetry in four cases — 



(ca) gives us 8 E ; 

 (bd) „ 8 G; 



(fa) gives us 8 A ; 



(fc) » 8 B; 



W . 8 C; 



(fd) „ 8 D; 



whose symmetry and circles are read on their figures, where the zoneless poles 

 on 8 A, 8 B, and 8 D are 55 and 33, 44 and 33, 44 and 44, and the leading flaps 

 are marked 78. 



42. We take next 6 B, whose only faces are the 2-zoned polar 1256, and the 

 monozone faces 4253 and 124. The only lines drawable 



are, 



fa, ba, lib, each 35, 



ff, aa, bb, ha, each =44, 

 and f'b, bb, each =43. 



Here (43) is fixed for every flap that we can draw, 



[The 6 should bo at the base corner.] ^^ ^ ^ ^ each = 44. 



(ba) = 53, 54>(56)=53, 54? (12) and (43) are fixed (arts. 22, 23). 



(hb)= 53>(12)=44and >(56)=43. 



(66) =44, 44>(12) or (56) or (43) ? each=44,44. 



(/)=44,33>(43)=44,33? 



(aa)=44 >(12)=43 and >(56)=43. 



